Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\tan \left(x-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The period is \(\pi\), and asymptotes are at \(x = \frac{3\pi}{4} + k\pi\).

Step 1: Identify the Basic Period of Tangent

The basic period of the tangent function, \(y = \tan x\), is \(\pi\). This means that the function repeats its values every \(\pi\) units along the x-axis.

Step 2: Analyze the Transformation in the Function

The given function is \(y = \tan \left(x - \frac{\pi}{4}\right)\). The term \(x - \frac{\pi}{4}\) represents a horizontal shift of \(\frac{\pi}{4}\) units to the right. This does not affect the period, so the period remains \(\pi\).

Step 3: Determine the Asymptotes

Tangent functions have vertical asymptotes where the function is undefined. For the general tangent function \(y = \tan (x)\), these occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. After the horizontal shift, the asymptotes become \(x = \frac{\pi}{2} + k\pi + \frac{\pi}{4}\). Simplifying, they are at \(x = \frac{3\pi}{4} + k\pi\).

Step 4: Sketch the Graph

1. Draw the x-axis and y-axis. 2. Mark the asymptotes at \(x = \frac{3\pi}{4} + k\pi\).3. Draw the basic shape of one period of the tangent function between \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\), crossing the x-axis at \(x = \frac{5\pi}{4}\).4. Repeat this shape to the left and right along the x-axis, considering the periodicity of \(\pi\).

Key Concepts

Graph Asymptotes

In the world of trigonometric functions, asymptotes are crucial elements to understand. For the tangent function, these are the lines that the graph approaches but never touches, marking points of discontinuity.
They occur where the tangent function becomes undefined. For a standard tangent function, asymptotes occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is any integer.

• For the function \(y = \tan(x - \frac{\pi}{4})\), the graph shifts horizontally due to \(-\frac{\pi}{4}\).
• This shift means the asymptotes also shift by \(\frac{\pi}{4}\) to the right.
• Thus, our new asymptotes are at \(x = \frac{3\pi}{4} + k\pi\).

Understanding where these asymptotes lie helps in accurately sketching the graph, making sure the proper boundaries are set for each period of the tangent function.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions. The tangent function, \(y = \tan x\), has a natural periodicity of \(\pi\).
This means that the graph pattern repeats itself every \(\pi\) units along the x-axis, creating a continuous wave that repeats indefinitely.

• When examining \(y = \tan(x - \frac{\pi}{4})\), observe that while it includes a horizontal shift, the periodicity remains unchanged.
• The period is still \(\pi\) since the horizontal transformation only shifts the entire graph to a new position without altering the distance over which the pattern repeats.

Understanding periodicity is essential when constructing a graph, as it reminds us where the function "starts over" and ensures each repeat segment mirrors the original segment within any set period.

Horizontal Shift

The horizontal shift is a transformation that moves the entire graph of a function left or right along the x-axis.
For \(y = \tan(x - \frac{\pi}{4})\), it's important to recognize how this affects the graph.

• The expression \(x - \frac{\pi}{4}\) indicates a shift of \(\frac{\pi}{4}\) units to the right.
• This movement doesn't alter the shape of the graph or the period, but it does move features, like zeros and asymptotes, over by \(\frac{\pi}{4}\) units.
• After shifting, points that were initially at \(x = 0\), \(x = \pi\), align now at \(x = \frac{\pi}{4}\), \(x = \pi + \frac{\pi}{4}\).

These shifts are fundamental when graphing since they determine where each element of the tangent function draws its pattern on the graph.

Trigonometric Graph Sketching

Trigonometric graph sketching involves plotting functions like tangent by hand, taking into account transformations such as shifts and periodicity.
Understanding how to sketch these graphs is a necessary skill for visualizing mathematical relationships.

• Start by marking the asymptotes at \(x = \frac{3\pi}{4} + k\pi\), creating clear boundaries for each repeating segment.
• Within these boundaries, draw the basic shape for a tangent function: rising from negative infinity, crossing the x-axis, and heading to positive infinity.
• For \(y = \tan(x - \frac{\pi}{4})\), observe that the zero, or crossing point, is now at \(x = \frac{5\pi}{4}\).
• Continue sketching the pattern over additional periods in both directions as \(k\) varies.

By breaking down the sketching into these steps and understanding each transformation's effect, you can accurately represent how the function behaves across its domain.